function [J, grad] = linearRegCostFunction(X, y, theta, lambda)
%LINEARREGCOSTFUNCTION Compute cost and gradient for regularized linear 
%regression with multiple variables
%   [J, grad] = LINEARREGCOSTFUNCTION(X, y, theta, lambda) computes the 
%   cost of using theta as the parameter for linear regression to fit the 
%   data points in X and y. Returns the cost in J and the gradient in grad

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;
grad = zeros(size(theta));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost and gradient of regularized linear 
%               regression for a particular choice of theta.
%
%               You should set J to the cost and grad to the gradient.
%

J0 = sum((X * theta - y).^2) ./ (2*m);
reg = lambda*sum(theta(2:length(theta)).^2)/(2*m);
J = J0 + reg;

grad_without_reg = X'*(X*theta - y)./m;
% no regression part for theta0
temp = theta;
temp(1) = 0;
derivative_reg = lambda.*temp./m;
grad = grad_without_reg + derivative_reg;

% =========================================================================

grad = grad(:);

end
